CHAPTER 09

Probability Foundations

Develop the probability ideas needed to understand uncertainty, random variables and stochastic models.

\[ \text{outcomes} \rightarrow \text{events} \rightarrow \text{probability} \rightarrow \text{random variables} \]

PROBABILITY

From uncertainty to random variables

Random Experiments

Understand situations where the possible outcomes are known but the actual outcome is uncertain.

Learn → 02

Outcomes and Events

Distinguish individual outcomes from collections of outcomes called events.

Learn → 03

Probability

Interpret probability as a numerical measure of uncertainty between \(0\) and \(1\).

Learn → 04

Conditional Probability

Update probabilities when additional information is known.

Learn → 05

Independence

Understand when knowledge of one event does not change the probability of another.

Learn → 06

Random Variables

Represent outcomes of a random experiment using numerical values.

Learn → 07

Discrete Distributions

Describe probabilities assigned to countable values of a random variable.

Learn → 08

Continuous Distributions

Describe probabilities using density functions over continuous ranges of values.

Learn → 09

Expectation

Understand the probability-weighted average value of a random variable.

Learn → 10

Variance and Standard Deviation

Measure how strongly random outcomes are spread around their expected value.

Learn → 11

Covariance and Correlation

Measure how two random quantities tend to vary together.

Learn → 12

Conditional Expectation

Understand expected values when some information about the system is already known.

Learn →

The central idea

\[ \mathbb{E}[X] = \sum_x x\,P(X=x) \]

Probability describes uncertainty. Random variables attach numbers to uncertain outcomes, allowing us to calculate quantities such as expectation and variance and eventually build stochastic models.